Staff: Mentor

I think that you are confusing yourself with your notation. The normal force is a vector, not a scalar. So you would not have N=mg/cos30, you would have Ny=mg. However, even that is not sufficient if there is friction as well as the normal force.

You need to break all three forces into components, so in vectors you have N+F+mg=0, and in components you would have Ny+Fy+mgy=0 and Nx+Fx+mgx=0 [where ##\mathbf{g}=(g_x,g_y)=(0,-|\mathbf{g}|)##, and ##\mathbf{N}=(N_x,N_y)=(-|\mathbf{N}|\sin 30,|\mathbf{N}|\cos 30)## and ##\mathbf{F}=(F_x,F_y)=(-|\mathbf{F}|\cos 30,|\mathbf{F}|\sin 30)##].

One of the reasons to use the ramp coordinate system is that you only need to break one vector (the weight) into components. Using the ground coordinate system you need to break both the normal and the friction force into components, and if there is any acceleration then you have to break that into components also.